## 607 Reputation

19 years, 249 days

## Sorry...

```We compute a system of PDE's for J[1](alpha x)

f:=BesselJ(1,alpha*x):

by first computing an ODE with respect to the variable x
using gfun[holexprtodiffeq] ,  and next considering symmetries of
f to derive a complete system.

ff:=holexprtodiffeq(f,y(x));

//          2  2\          / d      \    2 / d  / d      \\            1
{ \-1 + alpha  x / y(x) + x |--- y(x)| + x  |--- |--- y(x)||, D(y)(0) = - alpha
\                          \ dx     /      \ dx \ dx     //            2

\
}
/
diffeq:=op( select(has,ff,x) );

/          2  2\          / d      \    2 / d  / d      \\
\-1 + alpha  x / y(x) + x |--- y(x)| + x  |--- |--- y(x)||
\ dx     /      \ dx \ dx     //

diffeqx:=subs(y(x)=h(x,alpha),diffeq);

/          2  2\                 / d             \    2 / d  / d             \\
\-1 + alpha  x / h(x, alpha) + x |--- h(x, alpha)| + x  |--- |--- h(x, alpha)||
\ dx            /      \ dx \ dx            //

diffeqalpha:=subs({x=alpha,alpha=x,y(x)=h(x,alpha)},diffeq);

/          2  2\                     /   d               \
\-1 + alpha  x / h(x, alpha) + alpha |------- h(x, alpha)|
\ dalpha            /

2 /   d    /   d               \\
+ alpha  |------- |------- h(x, alpha)||
\ dalpha \ dalpha            //

sol:=pdsolve([diffeqx,diffeqalpha]);

{h(x, alpha) = _C1 BesselJ(1, alpha x) + _C2 BesselY(1, alpha x)}

diffeqxalpha:=x*D[1](h)(x,alpha)-alpha*D[2](h)(x,alpha);

x D[1](h)(x, alpha) - alpha D[2](h)(x, alpha)

solall:=pdsolve([diffeqx,diffeqalpha,diffeqxalpha]);

{h(x, alpha) = _C2 BesselJ(1, alpha x) + _C1 BesselY(1, alpha x)}
```

## Sorry...

```We compute a system of PDE's for J[1](alpha x)

f:=BesselJ(1,alpha*x):

by first computing an ODE with respect to the variable x
using gfun[holexprtodiffeq] ,  and next considering symmetries of
f to derive a complete system.

ff:=holexprtodiffeq(f,y(x));

//          2  2\          / d      \    2 / d  / d      \\            1
{ \-1 + alpha  x / y(x) + x |--- y(x)| + x  |--- |--- y(x)||, D(y)(0) = - alpha
\                          \ dx     /      \ dx \ dx     //            2

\
}
/
diffeq:=op( select(has,ff,x) );

/          2  2\          / d      \    2 / d  / d      \\
\-1 + alpha  x / y(x) + x |--- y(x)| + x  |--- |--- y(x)||
\ dx     /      \ dx \ dx     //

diffeqx:=subs(y(x)=h(x,alpha),diffeq);

/          2  2\                 / d             \    2 / d  / d             \\
\-1 + alpha  x / h(x, alpha) + x |--- h(x, alpha)| + x  |--- |--- h(x, alpha)||
\ dx            /      \ dx \ dx            //

diffeqalpha:=subs({x=alpha,alpha=x,y(x)=h(x,alpha)},diffeq);

/          2  2\                     /   d               \
\-1 + alpha  x / h(x, alpha) + alpha |------- h(x, alpha)|
\ dalpha            /

2 /   d    /   d               \\
+ alpha  |------- |------- h(x, alpha)||
\ dalpha \ dalpha            //

sol:=pdsolve([diffeqx,diffeqalpha]);

{h(x, alpha) = _C1 BesselJ(1, alpha x) + _C2 BesselY(1, alpha x)}

diffeqxalpha:=x*D[1](h)(x,alpha)-alpha*D[2](h)(x,alpha);

x D[1](h)(x, alpha) - alpha D[2](h)(x, alpha)

solall:=pdsolve([diffeqx,diffeqalpha,diffeqxalpha]);

{h(x, alpha) = _C2 BesselJ(1, alpha x) + _C1 BesselY(1, alpha x)}
```

## PDE system for Bessel...

I see what you say, although I don't know the Lie algebraic methods for pdes. In recent case however the solution sets are the same. f:=BesselJ(1,alpha*x): ff:=holexprtodiffeq(f,y(x)); print({(-1+alpha^2*x^2)*y(x)+x*(diff(y(x), x))+x^2*(diff(y(x), x, x)), (D(y))(0) = (1/2)*alpha}); // 2 2\ / d \ 2 / d / d \\ 1 \ { \-1 + alpha x / y(x) + x |--- y(x)| + x |--- |--- y(x)||, D(y)(0) = - alpha } \ \ dx / \ dx \ dx // 2 / diffeq:=op( select(has,ff,x) ); / 2 2\ / d \ 2 / d / d \\ \-1 + alpha x / y(x) + x |--- y(x)| + x |--- |--- y(x)|| \ dx / \ dx \ dx // diffeqx:=subs(y(x)=h(x,alpha),diffeq); / 2 2\ / d \ 2 / d / d \\ \-1 + alpha x / h(x, alpha) + x |--- h(x, alpha)| + x |--- |--- h(x, alpha)|| \ dx / \ dx \ dx // diffeqalpha:=subs({x=alpha,alpha=x,y(x)=h(x,alpha)},diffeq); / 2 2\ / d \ \-1 + alpha x / h(x, alpha) + alpha |------- h(x, alpha)| \ dalpha / 2 / d / d \\ + alpha |------- |------- h(x, alpha)|| \ dalpha \ dalpha // sol:=pdsolve([diffeqx,diffeqalpha]); {h(x, alpha) = _C1 BesselJ(1, alpha x) + _C2 BesselY(1, alpha x)} diffeqxalpha:=x*D[1](h)(x,alpha)-alpha*D[2](h)(x,alpha); x D[1](h)(x, alpha) - alpha D[2](h)(x, alpha) solall:=pdsolve([diffeqx,diffeqalpha,diffeqxalpha]); {h(x, alpha) = _C2 BesselJ(1, alpha x) + _C1 BesselY(1, alpha x)}

## PDE system for Bessel...

I see what you say, although I don't know the Lie algebraic methods for pdes. In recent case however the solution sets are the same. f:=BesselJ(1,alpha*x): ff:=holexprtodiffeq(f,y(x)); print({(-1+alpha^2*x^2)*y(x)+x*(diff(y(x), x))+x^2*(diff(y(x), x, x)), (D(y))(0) = (1/2)*alpha}); // 2 2\ / d \ 2 / d / d \\ 1 \ { \-1 + alpha x / y(x) + x |--- y(x)| + x |--- |--- y(x)||, D(y)(0) = - alpha } \ \ dx / \ dx \ dx // 2 / diffeq:=op( select(has,ff,x) ); / 2 2\ / d \ 2 / d / d \\ \-1 + alpha x / y(x) + x |--- y(x)| + x |--- |--- y(x)|| \ dx / \ dx \ dx // diffeqx:=subs(y(x)=h(x,alpha),diffeq); / 2 2\ / d \ 2 / d / d \\ \-1 + alpha x / h(x, alpha) + x |--- h(x, alpha)| + x |--- |--- h(x, alpha)|| \ dx / \ dx \ dx // diffeqalpha:=subs({x=alpha,alpha=x,y(x)=h(x,alpha)},diffeq); / 2 2\ / d \ \-1 + alpha x / h(x, alpha) + alpha |------- h(x, alpha)| \ dalpha / 2 / d / d \\ + alpha |------- |------- h(x, alpha)|| \ dalpha \ dalpha // sol:=pdsolve([diffeqx,diffeqalpha]); {h(x, alpha) = _C1 BesselJ(1, alpha x) + _C2 BesselY(1, alpha x)} diffeqxalpha:=x*D[1](h)(x,alpha)-alpha*D[2](h)(x,alpha); x D[1](h)(x, alpha) - alpha D[2](h)(x, alpha) solall:=pdsolve([diffeqx,diffeqalpha,diffeqxalpha]); {h(x, alpha) = _C2 BesselJ(1, alpha x) + _C1 BesselY(1, alpha x)}

## Bessel...

I see. Thanks. I began to rework the original worksheet. The PDE of x is ok. The PDE of BesselJ(1,a*x) is problematical. I have a partial diff eq system subs(y(x)=h(x,a),deq); and subs({x=a,a=x,y(x)=h(x,a)},deq); It's ok. But what is the meaning of the line x(D_x h(x,a)) - a(D_a h(x,a)) ? Why don't they take the pde system sys2:={ x^2*(D[1,1](h)(x,a))+x*(D[1](h)(x,a))+(-1+a^2*x^2), a^2*(D[2,2](h)(x,a))+a*(D[2](h)(x,a))+(-1+a^2*x^2)};

## Bessel...

I see. Thanks. I began to rework the original worksheet. The PDE of x is ok. The PDE of BesselJ(1,a*x) is problematical. I have a partial diff eq system subs(y(x)=h(x,a),deq); and subs({x=a,a=x,y(x)=h(x,a)},deq); It's ok. But what is the meaning of the line x(D_x h(x,a)) - a(D_a h(x,a)) ? Why don't they take the pde system sys2:={ x^2*(D[1,1](h)(x,a))+x*(D[1](h)(x,a))+(-1+a^2*x^2), a^2*(D[2,2](h)(x,a))+a*(D[2](h)(x,a))+(-1+a^2*x^2)};

## Four Bessel...

I also guessed that h(x,a)=x. :-) sys1:={ x*(d/dx h(x,a)) -1, d/da h(x,a) } Then d/da h(x,a)=0. Right. But x*(d/dx h(x,a)) -1 not=0. What is zero, is d/dx h(x,a) -1 So I think this worksheet is not only too outdated, but there is a lot of error that probably can be corrected. However, the idea is clear and when it needed I will be able to apply.

## Four Bessel...

I also guessed that h(x,a)=x. :-) sys1:={ x*(d/dx h(x,a)) -1, d/da h(x,a) } Then d/da h(x,a)=0. Right. But x*(d/dx h(x,a)) -1 not=0. What is zero, is d/dx h(x,a) -1 So I think this worksheet is not only too outdated, but there is a lot of error that probably can be corrected. However, the idea is clear and when it needed I will be able to apply.

## Bessel...

Many thanks. Recently I'm working on Bessel functions and any help is appreciated. Which packages you recommend to use?

## Bessel...

Many thanks. Recently I'm working on Bessel functions and any help is appreciated. Which packages you recommend to use?

## Bessel...

The origin of my question can be found here http://algo.inria.fr/libraries/autocomb/FourBessel.mws http://algo.inria.fr/libraries/autocomb/FourBessel.ps http://algo.inria.fr/libraries/autocomb/FourBessel-html/FourBessel.html They denote by kappa the integral in my question. To tell the truth I don't understand the very first step, namely, The identity function x trivially satisfies the following differential system > sys1:={ x*(d/dx h(x,a)) -1, d/da h(x,a) } where each entry expr in the set denotes the equation expr=0. What does h(x,a) mean? x? Then the first equation is wrong.

## Bessel...

The origin of my question can be found here http://algo.inria.fr/libraries/autocomb/FourBessel.mws http://algo.inria.fr/libraries/autocomb/FourBessel.ps http://algo.inria.fr/libraries/autocomb/FourBessel-html/FourBessel.html They denote by kappa the integral in my question. To tell the truth I don't understand the very first step, namely, The identity function x trivially satisfies the following differential system > sys1:={ x*(d/dx h(x,a)) -1, d/da h(x,a) } where each entry expr in the set denotes the equation expr=0. What does h(x,a) mean? x? Then the first equation is wrong.

## Bessel functions...

Dear JacquesC, Axel, This solution is indeed very nice, unfortunately I don't know Mellin transformation, so I will learn its basic ideas. That paper is very interesting. Hopefully in the near future in Maple xx it will be involved. Many thanks.

## Bessel functions...

Dear JacquesC, Axel, This solution is indeed very nice, unfortunately I don't know Mellin transformation, so I will learn its basic ideas. That paper is very interesting. Hopefully in the near future in Maple xx it will be involved. Many thanks.

## RootFinding Isolate,Next...

The Next is very efficient. My proc is very fast. Thanks again. Sandor
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